Exercises - Moral hazard

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1 Exercises - Moral hazard 1. (from Rasmusen) If a salesman exerts high e ort, he will sell a supercomputer this year with probability 0:9. If he exerts low e ort, he will succeed with probability 0:5. The company will obtain a revenue of 2000 if the sale is made. The salesman would require a wage of 50 if he had to exert low e ort, but 70 if he had to exert high e ort, he is risk neutral, and his utility is separable in e ort and money. That, is U (w; e) = w v (e), where w and e denote the wage and the e ort level, respectively. The preferences of the company are represented by B (w; x) = x w where x is the revenue. The preferences of the two participants satisfy the assumptions of von Neuman-Morgenstern. (a) Prove that high e ort is rst-best e cient. (b) How high would the probability of success with low e ort have to be for high e ort to be ine cient? (c) If you cannot monitor the programmer and cannot pay him a wage contingent on success, what should you do? (d) Now, suppose you can make the wage contingent on success. Let the wage be w e if he makes a sale and w ne if he does not. w e and w ne will have to satisfy two conditions: a participation constraint and an incentive compatibility constraint. What are they? (e) What is a contract that will achieve the rst best? (f) Now suppose the salesman is risk averse, and his utility from money is ln w. Set up the participation and incentive compatibility constraints again and nd the optimal contract. i. Does the expected payment of the rm in the optimal contract rise, fall, or stay the same, compared with the situation where the salesman is risk-neutral? ii. Does the gap between w e and w ne rise, fall, or stay the same, compared with the situation where the salesman is risk-neutral? 2. A Principal (P )considers the possibility of hiring a worker (A). The worker can exert two e ort levels and productivity of the worker is summarized in the next table of conditional probabilities. x 1 x 2 x 3 e 1 0:5 0:3 0:2 e 2 0:2 0:4 0:4 The preferences over certain consumptions are represented by the elementary utility functions B (x w) = ln(x w) for P, and U (w; e) = u (w) v (e) where u (w) = w for A. 1

2 (a) Knowing that the preferences of both P and A over lotteries satisfy the axioms of von Neumann-Morgenstern, how can we represent these preferences? (b) Is there any stochastic dominance relationship between the lotteries generated by the two di erent e ort levels? (c) Assume that x 1 = 10; x 2 = 100; x 3 = 1000, v (e 1 ) = 0, v (e 2 ) = 10 and that the reservation utility of the agent is U = 50. Calculate (and explain) the contract that the principal will o er to the agent. i. when there is symmetric information. ii. when there is asymmetric information and P cannot observe e. (d) Find (and explain) the certainty equivalent of any the contracts for P and A. 3. The next table summarizes the conditional probabilities generated from the bilateral relationship between an agent (A) and a principal (P ) x 1 = 40 x 2 = 100 e 1 0:5 0:5 e 2 0:2 0:8 P is risk-neutral, with B (x w) = x w, and the elementary utility of A is given by u (w) = w 1=2, v (e 1 ) = 0 and v (e 2 ) = c. (a) Find the (b) Fins the optimal contract of P when there is asymmetric information (consider w i 0). (c) What are the cost of the asymmetric information? (it is a general question and the exercise is just an example). 4. The next table summarizes the conditional probabilities generated from the bilateral relationship between an agent (A) and a principal (P ) x 1 = 1 x 2 = 2 x 3 = 4 x 4 = 6 x 5 = 10 x 6 = 20 x 7 = 60 e 1 = 0 0:25 0:2 0:05 0:05 0:1 0:15 0:2 e 2 = 1 0:15 0:1 0:15 0:05 0:15 0:2 0:2 Assuming that P is risk-neutral and A is risk-averse, with an elementary utility function which is separable in w and e, nd the relationship among the wages obtained by A when the principal induces e = e 2 and there is asymmetric information. Explain. 5. (de Marcos Vera Hernández) Consider a seller with elementary utility function u (w; e) = w 0:5 e 2, where w denotes the wage and e is his e ort level. He can choose between e = 0 and e = 3, and his reservation utility 2

3 is 21. There are three possible levels of sales: 0; 1000 and The conditional probabilities are summarized in the next table: x 1 = 0 x 2 = 1000 x 3 = 2500 e = 0 0:4 0:4 0:2 e = 3 0:2 0:4 0:4 The owner maximizes her expected bene ts (revenue minus wages). The preferences of the two participants satisfy the assumptions of von Neuman- Morgenstern. (a) Write the optimization problems of P assuming that both sales and e ort are observable by P. (b) Solve these problems and nd the optimal contract and nd the optimal contract when the e ort level is observed. (c) Write the optimization problem of the owner when she wants to induce e = 3 assuming that she cannot observe this e ort level (sales are). (d) Solve the above problem (you can assume w i 0). (e) Find the optimal contract that will be o ered by P when e is not observable. 6. Consider a principal (P) deciding whether to hire an agent (A). The reservation utility of A is U = 0, and his elementary utility depends on the wage w and the e ort level e and is given by U (w; e) = u (w) v (e), where u (w) = ln w and v (e) = e 2. Assume there are two possible e ort levels e 2 fe a ; e b g f2; 0g and two possible outcomes x 2 fx 1 ; x 2 g f0; 100g. The conditional probabilities of these outcomes are summarized next: e a = 2 e b = 0 x 1 = x 2 = The principal is risk-neutral and her elementary utility function is B (w; x) = x w (x; e), where w (x; e) is the wage. (a) Find the contract that P will o er to the agent if there is symmetric information. (b) Find the contract that P will o er to the agent when she only observes the outcome, but not the e ort level Explain the di erences with the contract obtained in (a). 7. Consider a principal (P ) choosing the contract to be o ered to an agent (A) to perform some task. This agent can exert either low e ort (e) or 3

4 high e ort (e) to this task. In each case, the e ort level generates a lottery over the possible outcomes x 1 = 10 and x 2 = 90, which are summarized in the next table: x 1 x 2 e 3=5 2=5 e 2=5 3=5 The elementary utility functions are given by B(x i ; w i ) = (x i w i ) 1=2 and U (w i ; e) = w 1=2 i v (e) where v (e) = 9=5 and v (e) = 22=5, and the reservation utility of the agent is U = 0. (a) Find the contract that P will o er to the agent if there is symmetric information. Explain. (b) Find the certainty equivalent of any the contracts for P and A. Explain. (c) Explain how do the previous results change if the risk attitudes of the agents also change. 8. An agent has to perform a task for a Principal and he may exert e ort or to shirk. The following gives the probability of each outcome conditional on the level of e ort made by the agent. x 1 = 50 x 1 = 100 e = 0 0:6 0:4 e = 4 0:2 0:8 The principal is risk-neutral and the elementary utility function of the agent is given by u (w; e) = w e (a) Find the contract that P will o er to the agent if there is symmetric information. Explain. (b) Find the contract that P will o er to the agent when only the outcome is observed by the principal. Explain. (c) Explain the results obtained. 9. The following table summarizes the probabilities that the Principal obtains di erent revenues depending on the e ort made by an agent: x 1 = 0 x 2 = 10 x 3 = 50 e 1 1=3 1=3 1=3 e 2 1=2 0 1=2 e 3 1=4 1=4 1=2 4

5 The preferences of the two participants satisfy the assumptions of von Neuman-Morgenstern and can be represented by the following elementary utility functions B (x; w) = (x w) 1=2 U (w; e) = w e 2 The reservation utility of the agent is zero (a) Find the optimal contract for P if there is symmetric information. Explain. (b) Find the optima contract for P when there is moral hazard. (c) Discuss the possibilities of P in any of the previous cases. 10. A Principal P is deciding whether to hire an agent A to perform a task. This agent can This agent can engage in various e ort levels (e; e m ; e) for this task. In each case, a lottery over three possible results is obtained. Next table summarizes them: x 1 = 0 x 2 = 10 x 3 = 90 e 0:7 0:1 0:2 e m 0:6 0 0:4 e 0 0:4 0:6 The elementary utility function of P and A are given, respectively, by B(x i ; w i ) = x i w i and U (w i ; e) = ln w i v (e) where v (e) = 0; v (e m ) = ln 5 and v (e) = ln 10, and the reservation utility of the agent is U = 0. (a) Find the optimal contract for P if there is symmetric information. Explain. (b) Find the optimal contract for P when there is moral hazard. Explain. 11. A Principal P is deciding whether to hire an agent A to perform a task. The agent has a reservation utility of U. The elementary utility of the agent depends on the wage w and the e ort level e, and it is given by U (w; e) = u (w) v (e), where u 0 > 0; u 00 0 and v 0 > 0; v Assume the agent can engage in various e ort levels e 2 e L ; e H f0; 5g for this task, and three possible outcomes x 2 fx 1 ; x 2 ; x 3 g f0; 100; 400g can be obtained. The conditional probabilities are summarized next x 1 = 0 x 2 = 100 x 3 = 400 e L 0:6 0:3 0:1 e H 0:1 0:3 0:6 The principal is risk-neutral and her elementary utility function B (w; x) = x w (x; e) where w (x; e) denotes the wage received by the agent. 5

6 (a) Find the Principals optimal contract when she cannot observe the e ort level of the agent when u (w) = w, v (e) = e 2 and U = 81. (b) What would change if P could observe the e ort level? (c) Assume now u (w) = w 1=2, v (e) = e and U = 9. Find the optimal contract both when the e ort is not observable and when it is observable. Determine the (expected) e ciency loss due to asymmetric information. (d) Assume now that there is a minimum wage of 81 in the economy. How would this a ect the optimal contract? 12. A Principal (P ) is deciding to hire an agent (A). The reservation utility of A is U = 0 and his elementary utility function, which depends on the wage w and the e ort level e, is given by U (w; e) = u (w) v (e), where u (w) = ln w and v (e) = e 2. Assume there are three possible e ort levels e 2 fe a ; e m ; e b g f2; 1; 0g that can be exercised by the agent, and two possible outcomes x 2 fx 1 ; x 2 g f0; 100g. Next table summarizes the conditional probabilities over these outcomes: e a = 2 e m = 1 e b = 0 x 1 = 0 0:4 0:5 1 x 2 = 100 0:6 0:5 0 The principal is risk-neutral and her elementary utility function B (w; x) = x w (x; e) where w (x; e) denotes the wage received by the agent. (a) Find the Principals optimal contract when she can observe the e ort level of the agent. (b) Find the Principals optimal contract when she cannot observe the e ort level of the agent. Can she use arbitrarily large punishment threats in any case? 13. A rm is deciding to hire a worker to produce a good. The rm cannot observe the e ort level of the worker e, which will a ect the probability of obtaining each of the three possible results 5; 10 or Next table summarizes these conditional probabilities: e = 0 1=2 1=4 1=4 e = 1 1=4 1=2 1=4 If the rm is risk-neutral and the worker is risk-averse, explain how would be the optimal contracts o ered by the rm. 6

7 14. A rm is deciding to hire a worker to produce a good. Production can take two values 5 or 10 with probabilities that depend on the e ort level exerted by the worker, summarized next: 5 10 e = 0 1=2 1=2 e = 1 1=4 3=4 Assume that the elementary utility of the rm and the worker are given by B (x; w) = x w and U (w; e) = w 1=2 e, respectively, where w is the reward received by the worker and e is his e ort level. The reservation utility of the worker is U = 1. (a) Find the Principals optimal contract when she can observe the e ort level of the agent. Explain (b) Find the Principals optimal contract when she cannot observe the e ort level of the agent. Explain. 15. When an Agent is hired by a Principal and he chooses action A then he generates a lottery l A = (10; 100; 1=2; 1=2), while by choosing action B, the lottery is l B = (10; 100; 1=4; 3=4). The principal must o er a contract to this agent. The Principal id risk-neutral and the elementary utility of the agents is given by u (w; A) = w 1=2 and u (w; B) = w 1=2 1, depending on the action he chooses. His reservation utility is U = 3. (a) Find the Principals optimal contract when she can observe the e ort level of the agent. Explain (b) Find the Principals optimal contract when she cannot observe the e ort level of the agent. Explain. 16. Next table summarizes the probabilities for the Principal to obtain di erent outcomes as a function of the e ort e made by an Agent. x 1 = 0 x 2 = 10 x 3 = 50 e 1 = 4 1=4 1=2 1=4 e 2 = 6 1=4 1=4 1=2 The preferences of the Principal and the agent are represented, respectively, by the following elementary utility functions B (x; w) = (x w) 1=2 U (w; e) = w 1=2 Find the contract that the principal will o er to the agent when e ort is observable and the reservation utility of the agent is U. e 7

8 17. Next table summarizes the probabilities for the Principal to obtain di erent outcomes as a function of the e ort e made by an Agent. x 1 = 0 x 2 = 10 x 3 = 50 e 1 = 0 1=4 1=2 1=4 e 2 = 4 1=4 1=4 1=2 The preferences of the Principal and the agent are represented, respectively, by the following elementary utility functions B (x; w) = x U (w; e) = w w e Find the contract that the principal will o er to the agent when e ort is not observable and the reservation utility of the agent is U. 18. Next table summarizes the probabilities for the Principal to obtain di erent outcomes as a function of the e ort e made by an Agent. x 1 = 10 x 2 = 50 e 1 1=2 1=2 e 2 1=4 3=4 The preferences of the Principal and the agent are represented, respectively, by the following elementary utility functions B (x; w) = (x w) 1=2 U (w; e) = w Find the contract that the principal will o er to the agent when e ort is not observable and the reservation utility of the agent is U. Compare the result with the contract that (you think) will be obtained when the e ort level is observed by the principal. 19. Next table summarizes the probabilities for the Principal to obtain di erent outcomes as a function of the e ort e made by an Agent. x 1 = 10 x 2 = 50 e 1 = 0 1=2 1=2 e 2 = 4 1=4 3=4 The preferences of the Principal and the agent are represented, respectively, by the following elementary utility functions e B (x; w) = x U (w; e) = w 1=2 w e Find the contract that the principal will o er to the agent when e ort is not observable and the reservation utility of the agent is U. Compare the result with the contract that (you think) will be obtained when the e ort level is observed by the principal. 8

9 20. (from Inés Macho and David Perez) A Principal can hire an agent, whose e ort level determines an outcome x. Assume that uncertainty is represented by three possible states of the nature " 1 ; " 2 and " 3. When accepting a contract, the agent can choose between two e ort levels, e 1 and e 2 ). The outcome obtained can be either 30:000 or 60:000. Next table shows the dependence of the result as a function of the e ort and the state of the nature: " 1 " 2 " 3 e 1 = 4 60:000 60:000 30:000 e 2 = 6 30:000 60:000 30:000 Both, the Principal and the Agent believe that the probability of each state is 1=3. The preferences of the Principal and the agent are represented, respectively, by the following elementary utility functions B (x; w) = x w and U (w; e) = w 1=2 e 2 where x = x (e; ") is the (monetary) value of the outcome and w (which can depend on the outcome) represents the monetary payment received by the Agent. The reservation utility of the Agent is 114. (a) What is the e ort level and the payment scheme in a situation with symmetric information? what would happen in the Principal was not risk-averse? (b) Explain the problem faced by the Principal when there is asymmetric information and she can only observe the nal outcome. What payment schemes allow her to induce e ort e 1? What payment schemes allow her to induce e ort e 2? Find the optimal payment scheme that would be nally o ered to the agent. Discuss the results 21. Next table summarizes the probabilities for the Principal to obtain di erent outcomes as a function of the e ort e made by an Agent. x 1 = 10 x 2 = 50 e = e = 2 1=4 3=4 The preferences of the Principal and the agent are represented, respectively, by the following elementary utility functions B (x; w) = (x w) 1=2 The reservation utility of the agent is U = 0. U (w; e) = w e (1) Find the contract that the principal will o er to the agent when e ort is not observable. Compare the result with the contract that will be obtained when the e ort level is observed by the principal. 9

10 22. Exercises 2,3,6,7,9,11,12 form chapter 3 of the book "An introduction to the Economics of Information" (Inés Macho and David Pérez). 23. "Ejercicios 2" from http : ==idea:uab:es=dperez=teaching:html. 10